Quantum Models of Crystals

Typical models for illustrating the crystalline quantum states characterization will be exposed, in terms of reciprocal lattice, as following the present discussion follows (Putz, 2006). 

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Dealing with quantum models of crystals viewed as related yet distinct levels of approximations for electronic evolution/nature in the periodic lattice of a crystal stmcture ... 

Chapter 3 Quantum Roots of Crystals and Solids) The quantum mechanics postulates are shortly reviewed for their application in providing the basic crystal Bloch theorem, further specialization to the quantum modeling of crystals in reciprocal space, on various levels of electronic behavior from free to quasi-free, to quasi-binding, to tight-binding models. 

S.4.2.2 Quantum Model of Quasi-Free Electrons in Crystals... 

FIGURE 3.16 Energetic discretization at the frontier of the first Brillouin zone in the quantum model of quasi-free electrons in crystal after (Further Readings on Quantum Solid 1936-1967 Putz,2006). 

Quantum Model of Tight-Binding Electrons in Crystal... 

A quantitative quantum approach will be presented in the next section. 2.4.2 THE QUANTUM MODEL OF THE CRYSTAL FIELD... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

After nearly two decades of controversy, it is now generally accepted that helium crystals are model systems for the general study of crystal surfaces, but also exceptional in having unique quantum properties. Helium crystals exhibit faceting as do ordinary crystals, but no other crystals can grow and melt sufficiently fast to make the propagation of crystallization waves possible at their surfaces. 

P. Bienstman and R. Baets, Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers, Opt. Quantum Electron. 33, 327-341 (2001). 

Static dielectric measurements [8] show that all crystals in the family exhibit a very large quantum effect of isotope replacement H D on the critical temperature. This effect can be exemphfied by the fact that Tc = 122 K in KDP and Tc = 229 K in KD2PO4 or DKDP. KDP exhibits a weak first-order phase transition, whereas the first-order character of phase transition in DKDP is more pronounced. The effect of isotope replacement is also observed for the saturated (near T = 0 K) spontaneous polarization, Pg, which has the value Ps = 5.0 xC cm in KDP and Ps = 6.2 xC cm in DKDP. As can be expected for a ferroelectric phase transition, a decrease in the temperature toward Tc in the PE phase causes a critical increase in longitudinal dielectric constant (along the c-axis) in KDP and DKDP. This increase follows the Curie-Weiss law. Sc = C/(T - Ti), and an isotope effect is observed not only for the Curie-Weiss temperature, Ti Tc, but also for the Curie constant C (C = 3000 K in KDP and C = 4000 K in DKDP). Isotope effects on the quantities Tc, P, and C were successfully explained within the proton-tunneling model as a consequence of different tunneling frequencies of H and D atoms. However, this model can hardly reproduce the Curie-Weiss law for Sc-... 

Different crystal sizes (some in the quantum size regime) were obtained by varying the film thickness. Three interconnected PL peaks at ca. 1.83,1.35, and 1.06 eV were obtained (no green emission) [32], A model of transitions from a deep donor (Cd-0 complex) level to various other levels was suggested to explain these peaks. 

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